"Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." - Bertrand Russell
We all have that friend(or maybe you're that person) who once said, "What's the use of mathematics? I'm never going to use this in life." But it turns out if you want to live a decent human life, you need mathematics more than you need food. The human brain has evolved in such a way that it is able to run about 38 thousand trillion (38 000 000 000 000 000) operations per second. Some of these operations are quite obvious, you want to cross the road and a car is coming. In less than a second, you're brain does a rough calculation estimating the relative speed of that car to you, and whether it will be safe to cross. Or when you see your crush, your brain will analyse their emotions and will run dozens of simulations for you to decide whether it is the right time for you to talk to them. But the majority of the calculations that your brain makes, are in the subconscious. When you saw that car, your nervous system sent a signal to your iris to increase your vision. And when you saw your crush, your brain was analysing whether they were a suitable mate and whether the love hormone, oxytocin, should be released. Now we will have that friend of ours tell us that, "Sure, the brain is mathematical. So what? I can use a computer without knowing any programming."
There is no clear definition of mathematics but we can all agree that it studies patterns, comes out with certain rules, which within certain axioms, stand as absolute truth. For example, we see the pattern that if one finger represents one apple and a second finger represents a second apple then one finger can also represent one orange and a second finger can represent a second orange. If within our axiom, oranges are not equal to apples, five apples plus five oranges will never give us ten oranges or ten apples. But five apples and another five apple will give us ten apples. And that it is an absolute truth. The work of mathematics goes beyond comparing apples and oranges and counting them. In fact, I'll argue that you need mathematics to solve any problem.
From Roman architecture to the paintings of the Renaissance, artists have been able to use the patterns which are appealing to the human senses. It is still a mystery why we like some patterns more than others or why we like symmetry so much. But without great mathematics, there could never be great art. The artists were certainly not solving equations, but they were able to express the right patterns in their work. How could a deaf Beethoven compose the Moonlight Sonata? He knew the structure of what makes great music and wrote it. Likewise, the Shakespeare's Sonnets are beautiful not only because of the words but because of the structure.
Mathematics does not make only art better but also our daily lives. It helps us deal with other people. Sometimes we are in direct competition with others. And in order for us to come to the best winning strategy, we need to use mathematics, actually a field of mathematics, called Game Theory. It was taken for granted since Adam Smith published The Wealth of Nations in 1776 that free market capitalism was the right economic system. Capitalism was fundamental to the rise of the British Empire and The United States of America. But it has also led to catastrophes like the Great Depression of 1929. Adam Smith was not wrong but his work lacked the maths. He did have the data to support his hypothesis but he did not really consider the way that individuals interact with one another in order to come up with an equilibrium where some people would win but nobody would lose. It was only in 1950 that American mathematician, John Nash came up with a mathematical supplement to Adam Smith's theory. It is now called the Nash equilibrium. Today, Game theory is used in politics and in biology to help solve problems without creating new ones.
(excerpt from the movie "A beautiful mind" about John Nash where the Nash equilibrium is explained)
It was thought for centuries that mathematics was something abstract and was not useful in knowing how the world works. Philosophy or religion was to explain how the universe works. Science was called natural philosophy. And then came Newton and he used mathematics to make predictions about how the universe works. And when his predictions turned out to be right, the place of mathematics in science was no longer questioned. At some point, mathematics diverged from physics. When German mathematician, Bernhard Riemann, set out the equations which described curved space, it did not have any practical applications. But in 1915, when Albert Einstein formulated the general theory of relativity, the geometry of curved space proved to be useful. Without the geometry of curved space, we would not be able to use GPS. Quantum mechanics introduced new mathematics to physics. And lately, string theory introduced new mathematics to physics. Physics explains how our universe works while mathematics explains how a universe works.
Let's come back to our friend that said that he would never use maths in their life. Well, they were not totally wrong. They were not taught real mathematics. Real mathematics is more than just adding numbers or solving equations. Mathematics can be expressed in shapes, in music, in architecture, in society and almost everywhere in the universe. Is the universe mathematical? Or is mathematics just a construction of our mind? It is hard to answer this question. But we can all agree that mathematics is the language in which the universe communicates with us. And if we analyse all our problems, we can always find the right mathematical tool to solve them. If not then we can always think of new tools. That's what mathematicians do.